250 DIELECTRICS. [ART, 480
If the difference of the potentials is kept unaltered, we substitute for p from equation (1). The force per unit of area is then (0 - a)2/87r (6 - t+t/K)'2.
479. A cylindrical dielectric. The outer and inner boundaries of two conductors A, B are infinite co-axial circular cylinders whose radii are a, b. A co-axial circular cylindrical dielectric C of specific inductive capacity JsTis introduced into the space between A and B, the rest of the space being rilled with air. To find the effect of the shell C on the capacity and potential energy.
Let />, p' be the densities of the charges on the surfaces A, B of the conductors ; ff, <r' those of the strata on C whose repulsions represent the forces due to the dielectric. Let the radii of the two surfaces L, L' of the shell C be a', b'; L being nearer A than B. Let a, ft be the potentials at the conductors, X, X' those at the surfaces L, I/.
The repulsion of any one of these cylinders at an internal point is zero. At an external point the force varies inversely as the distance r from the axis and is equal to 2m/r where m is the charge per unit of length (Arts. 55, 56). For the cylinder A,
The force at any point P between A and L is ^Trpajr. Putting r=a', and using the rule that the product of the force and K is unaltered when P passes into the dielectric (Art. 469), we see that the force just outside L is AirpajafK. The force at any point R between L and L' is therefore ^Tpajr'K where r' is the distance of 'R from the axis. Similarly the force at a point Q between L' and B is kirpajr" where r" is the distance of Q from the axis.
We now find by easy integrations
A i a' ^f\ 4nrpa, .6' ON/ * i &
X-a= -47rpalog~ , X'-X= -- —log—, B - X = - 4?rpa log —, .
a J\. a o
Adding these together we have
./}-.= -4Tp«(logJ + ltogJ + Iog^ ..................... (1).
The capacity C per unit of length (measured by the ratio of the charge on A to the difference of potentials) is given by ^7-,=log — ( 1 - •=. ) log 4 •
2t\j QJ \ It. J Cl
Since the whole quantity of matter given by Poisson's equivalent strata is zero (Art. 340), we have era' + cr'b' = 0. Also since the potential of the whole system at any point external to B is constant, the quantity (pa + p'b + cra' + cr'b')\()gr is independent of r, and this is impossible unless pa + pfb = Q. The' charges on the conductors A and B arc therefore equal and opposite.
The potential energy per unit of length (Art. 01) is given by
W— £ (2irpa a + 2irp'b p) = irpa (a - /3),
which can be expressed in terms of either the charge or the difference of potentials, by substituting from (1).
480. A repellwy point of mass E is placed at a point A in a medium of inductive capacity K ; prove that the potential at any point P distant r from A is EjKr.
The point may be regarded as the limit of a small sphere of equal mass, radius a, whose specific inductive capacity is unity.